What is the inverse of the function $f(x)=-\dfrac{1}{2}(x+3)$ ? $f^{-1}(x)=$
Explanation: Let's start by replacing $f(x)$ with $y$. $y=-\dfrac{1}{2}(x+3)$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=-\dfrac{1}{2}(x+3)$, so the inverse relationship is $x=-\dfrac{1}{2}(y+3)$. Solving this equation for $y$ will give us an expression for $f^{-1}(x)$. $\begin{aligned} x&=-\dfrac{1}{2}(y+3)\\\\ -2x&=y+3\\\\ -2x-3&=y\\\\\\ \end{aligned}$ The inverse of the function is $f^{-1}(x)=-2x-3$. [I saw someone solve this problem by originally solving for x. Were they wrong?]